Question

The current density in a cylindrical conductor of radius \(R\), varies as \(J(r)=J_{0} r / R\) (in the region from zero to \(R\) ). Express the magnitude of the magnetic field in the regions \(rR .\) Produce a sketch of the radial dependence, \(B(r)\)

Short answer

Based on the above step-by-step solution, the magnetic field B(r) inside a cylindrical conductor of radius R, with given current density J(r) = J0 * r / R, can be calculated using Ampere's Law. The magnetic field inside the conductor (r < R) is given by B(r) = (μ0J0r) / (2R) and outside the conductor (r > R) is given by B(r) = (μ0J0R²) / (2r). The radial dependence B(r) for the magnetic field is characterized by a linear increase with increasing radius inside the conductor and a 1/r decrease outside the conductor. The magnetic field reaches its maximum value at r = R and subsequently decreases with an increasing radius.

Step-by-step solution

Define a closed loop for Ampere's Law

In order to apply Ampere's Law, we need to choose an appropriate closed loop. We'll choose a circular loop with radius r centered at the conductor's axis. This loop is parallel to the cross-section of the conductor.

Write down Ampere's Law

Ampere's Law states that: \(\oint \vec{B} \cdot \vec{dL} = \mu_{0} I_{enclosed}\) where \(\oint \vec{B} \cdot \vec{dL}\) is the line integral of the magnetic field around the closed loop and \(I_{enclosed}\) is the current enclosed by the loop.

Solve for magnetic field inside the conductor (r < R)

For the loop inside the conductor, let us first find the \(I_{enclosed}\): \(dI = J(r) dA = J_0 \frac{r}{R} (2 \pi r dr)\) Integrate current density over the area enclosed by the loop: \(I_{enclosed} = \int_0^r J_0 \frac{r'}{R} (2 \pi r' dr') = \pi J_0 r^{2}\) Now, we can use Ampere's Law: \(\oint \vec{B} \cdot \vec{dL} = B(r) (2 \pi r) = \mu_{0} (\pi J_0 r^{2})\) Solve B(r) for r < R: \(B(r) = \frac{\mu_{0}J_{0}r}{2R}\) for \(r<R\)

Solve for magnetic field outside the conductor (r > R)

For the loop outside the conductor, the enclosed current is the total current I in the conductor: \(I_{enclosed} = \int_0^R J_0 \frac{r'}{R} (2 \pi r' dr') = \pi J_0 R^{2}\) Now, we can use Ampere's Law: \(\oint \vec{B} \cdot \vec{dL} = B(r) (2 \pi r) = \mu_{0} (\pi J_0 R^{2})\) Solve B(r) for r > R: \(B(r) = \frac{\mu_{0}J_{0}R^{2}}{2r}\) for \(r>R\)

Produce a sketch of the radial dependence, B(r)

Now, we can sketch the radial dependence B(r) derived in steps 3 and 4. The function B(r) for r < R shows a linear increase with r, while for r > R, B(r) decreases in a 1/r fashion. The magnetic field reaches its maximum value at r = R and then decreases with further increase in radius. The graph will have a steep positive slope inside the conductor and a steep negative slope outside the conductor, leading to a peak at r = R.

Key concepts

Ampere's Law

To grasp how a magnetic field behaves around a conductor, we first need to delve into Ampere's Law. This fundamental law of magnetism is akin to a magnetic version of Gauss's Law for electricity. It connects the magnetism around a current-carrying conductor to the current itself. In essence, Ampere's Law states that the line integral of the magnetic field \( \vec{B} \) around a closed loop is directly proportional to the electric current \( I_{enclosed} \) passing through any surface bounded by the loop.

Ampere's Law is mathematically expressed as:\
\[ \oint \vec{B} \cdot \vec{dL} = \mu_{0} I_{enclosed} \]
In this expression, \( \vec{dL} \) is an infinitesimal vector element of the closed loop, \( \mu_{0} \) is the permeability of free space, and \( I_{enclosed} \) is the current enclosed by the loop. This intricate dance between current and magnetic field allows us to solve for the magnetic field given a specific distribution of current.

Current Density

Current density, symbolized as \( J \), is a measure of how much current \( I \) flows through a unit area. It gives you a sense of how densely packed the current is within a conductor. The current density is particularly useful when the current is not uniformly distributed, as is the case in our cylindrical conductor scenario.

For our problem, the current density varies linearly with the radial distance \( r \) from the axis of the cylinder as:
\[ J(r) = J_{0} \frac{r}{R} \]
Here, \( J_{0} \) is the current density at the surface \( r = R \) of the cylinder. This linear relationship allows the intensity of the current to increase gradually from the center to the outer edge, and it significantly influences the resulting magnetic field inside the conductor.

Current Enclosed by Loop

To solve for the magnetic field inside our cylindrical conductor, we analyze the current enclosed by a hypothetical loop, which is necessary in applying Ampere's Law. The concept of 'current enclosed by a loop' refers to the total current passing through the area bounded by that loop. In our case, we're considering circular loops concentric with the cylinder.

For a loop of radius \( r \), using the given current density, we derive the enclosed current as:

• For \( r < R \) (inside the conductor), the enclosed current is the integral of the current density over the loop's area, producing \( I_{enclosed} = \pi J_0 r^{2} \).
• For \( r > R \) (outside the conductor), the enclosed current is the total current through the conductor's cross-sectional area, which is \( I_{enclosed} = \pi J_0 R^{2} \).

This distinction is crucial because it dictates the form of the magnetic field in different regions around the conductor.

Magnetic Field Sketch

A visual interpretation of the magnetic field strength as a function of distance from the conductor's center, or the magnetic field sketch, can provide intuitive understanding. To illustrate the behavior predicted by our findings, we sketch the magnetic field (B(r)) against the radial distance ( r ).

Inside the conductor \( (r < R) \), the magnetic field increases linearly with r due to the current density's linear relationship to r . Conversely, outside the conductor \( (r > R) \), it diminishes according to the inverse of the radial distance. The point where r = R marks the peak of the magnetic field, coinciding with the surface of the conductor. The sketch thus serves as a visual summary, revealing the increase in magnetic field up to the conductor's boundary and its decrease beyond that point.